The HVAC controller is tightly integrated into the entire IMA system. An issue with it will effect Autostop.

The challenge is that most Techs will not check the HVAC system for what are perceived as an engine/transmission issue.

You might want to visit a dealer explain the issue with autostop, and ask them to check if there are any codes stored in the HVAC module.

I had a similar situation with autostop and it took the dealer a week to figure it out. It is fixed now, and now my MPG is up 25%.